There are already many great booster designs out there - take for example the Saturn V, which has enough power to reasonably put almost anything into orbit.
Do new, developing space programs often use old, proven designs, or do they try to create their own booster from scratch?
It depends (when is that NOT the answer?). There are many examples of technology researched and/or developed in the past being reused in modern spaceflight.
SpaceX has explained that the prior work, available to them through NASA greatly decreased development time. (I recall references to the Apollo parachute design work, and the use of the basic PICA heat shield formulation, updated to modern times as PICA-X).
The Merlin engine uses a Pintle injector, used on the lunar module ascent engine.
Like all innovators, they stand upon the shoulders of those who preceded them.
However it is unclear that the resources made available to SpaceX, Orbital, Boeing, Lockheed-Martin, and Sierra Nevada Corp would be similarly made available to India, China or South Korea.
There is much in the public domain, as information, which might be available, but there are also more physical resources, (engine test stands, wind tunnels, research facilities) only available in the US that exist due to previous work by NASA. The Michoud facility that made the Saturn V first stage, reused some of the tooling to make the Shuttle external tank, and is now slated to make components for the SLS (Senate Launch System).
Conversely, look at the engines often employed by nascent space programs. Often they start with Solid rockets or else MMDH/MMH/Nitrogen Textroxide, which usually indicates that they have learned that LOX/LH engines are really hard to build. So start with something easier. Often, if they had a ballistic missile program, they will use that as the starting point for development.
The launch pads at Kennedy/Canaveral and Vandenberg AFB were developed at much cost, that the various launch providers in the US get access to for less than the real cost.
SpaceX was able to buy a used LOX storage ball that NASA was getting rid of, for very reduced pricing.
Sierra Nevada is using HL-20 research work for the design of their Dream Chaser vehicle.
Orbital seems to love reusing solid booster designs in their launchers (Minotaur, Taurus, Antares (upper stage), Pegasus, Pegasus II (For Stratolaunch)) that were developed for ICBM work.
So many examples of things, US companies can get access to, that would be very hard for anyone else to get access to. Which is very likely as it should be. (Return on investment?)
Not to discount the benefits of information, and the generally available information on engine design, cannot harm a developing space program. You can get amazing information on very specific details of the development of the F-1, SSME, and other engines that can only help anyone approaching the task. Some of the take away might be that it is 'very hard and expensive', from which SpaceX's approach of a simple design, scaled up through iterations might be the logical outcome.
